where \(a(\cdot , \cdot )\)  is the standard Galerkin FEM stiffness bilinear form, \(\eta \)  is the so-called softness parameter, and \(s(\cdot , \cdot )\)  is the bilinear form penalizing the gradient jumps across the mesh interfaces. The idea behind softFEM shares some common ground with isogeometric analysis where the basis functions have at least \(C^1\) -smoothness. In softFEM, the same basis functions are used as in Galerkin FEM so that the smoothness is only \(C^0\) . However, by considering the bilinear form \(\hat{a}(\cdot , \cdot )\)  instead of \(a(\cdot , \cdot )\) ,
one reduces the amount of energy stored in the gradient jumps of eigenfunctions associated with the large eigenvalues in the spectrum.
This change is not needed for eigenfunctions associated with the lower part of the spectrum since those eigenfunctions are smooth and can be accurately approximated on a given mesh.

We notice that the bilinear form \(s(\cdot , \cdot )\)  has been considered for the purpose of stabilization (i.e., leading to a positive contribution and not to a negative one as in the present work) in various contexts related, in particular, to advection-dominated advection-diffusion equations and to the Stokes equations [1]}, [2]}, [3]}.
In the context of the Helmholtz equation, the bilinear form
\(s(\cdot ,\cdot )\)  is weighted by a coefficient with positive imaginary part
to ensure coercivity [4]}. In addition, the possibility of using a weighting coefficient with negative real part has been considered in [5]}, [6]} to improve the phase error.
Incidentally, we mention that the term softFEM has been used recently in [7]} in a completely different context related to heuristic optimization and soft computing for solid mechanics.
<FIGURE>